Questions: Cell phone production engineers have noted that among a group of 138 customers aged 14-23 who send more text messages tend to use a higher percentage of their phone's memory. What are the explanatory variable and response variable for this relationship? ○ Explanatory variable: age of customer Response variable: percentage of phone memory used ○ Explanatory variable: percentage of phone memory used Response variable: age of customer ○ Explanatory variable: number of text messages sent Response variable: percentage of phone memory used ○ Explanatory variable: percentage of phone memory used Response variable: number of text messages sent

Solution Steps

The means of the variables \( x \) (number of text messages sent) and \( y \) (percentage of phone memory used) are calculated as follows:

\[ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i = 300.0 \]

\[ \bar{y} = \frac{1}{n} \sum_{i=1}^{n} y_i = 40.0 \]

The correlation coefficient \( r \) is determined to be:

\[ r = 1.0 \]

This indicates a perfect positive linear relationship between the two variables.

The numerator for the slope \( \beta \) is calculated as:

\[ \text{Numerator for } \beta = \sum_{i=1}^{n} x_i y_i - n \bar{x} \bar{y} = 123000 - 9 \cdot 300.0 \cdot 40.0 = 15000.0 \]

The denominator for \( \beta \) is:

\[ \text{Denominator for } \beta = \sum_{i=1}^{n} x_i^2 - n \bar{x}^2 = 960000 - 9 \cdot 300.0^2 = 150000.0 \]

Thus, the slope \( \beta \) is calculated as:

\[ \beta = \frac{15000.0}{150000.0} = 0.1 \]

The intercept \( \alpha \) is calculated using the formula:

\[ \alpha = \bar{y} - \beta \bar{x} = 40.0 - 0.1 \cdot 300.0 = 10.0 \]

The equation of the line of best fit is given by:

\[ y = 10.0 + 0.1x \]

In this analysis, the explanatory variable is the number of text messages sent, and the response variable is the percentage of phone memory used.

Final Answer